- Chapter 1: Equations and Inequalities
- Chapter 1.1: Graphs and Graphing Utilities
- Chapter 1.2: Linear Equations and Rational Equations
- Chapter 1.3: Models and Applications
- Chapter 1.4: Complex Numbers
- Chapter 1.5: Quadratic Equations
- Chapter 1.6: Other Types of Equations
- Chapter 1.7: Linear Inequalities and Absolute Value Inequalities
- Chapter 2: Functions and Graphs
- Chapter 2.1: Basics of Functions and Their Graphs
- Chapter 2.2: More on Functions and Their Graphs
- Chapter 2.3: Linear Functions and Slope
- Chapter 2.4: More on Slope
- Chapter 2.5: Transformations of Functions
- Chapter 2.6: Combinations of Functions; Composite Functions
- Chapter 2.7: Inverse Functions
- Chapter 2.8: Distance and Midpoint Formulas; Circles
- Chapter 3: Polynomial and Rational Functions
- Chapter 3.1: Quadratic Functions
- Chapter 3.2: Polynomial Functions and Their Graphs
- Chapter 3.3: Dividing Polynomials; Remainder and Factor Theorems
- Chapter 3.4: Zeros of Polynomial Functions
- Chapter 3.5: Rational Functions and Their Graphs
- Chapter 3.6: Polynomial and Rational Inequalities
- Chapter 3.7: Modeling Using Variation
- Chapter 4: Exponential and Logarithmic Functions
- Chapter 4.1: Exponential Functions
- Chapter 4.2: Logarithmic Functions
- Chapter 4.3: Properties of Logarithms
- Chapter 4.4: Exponential and Logarithmic Equations
- Chapter 4.5: Exponential Growth and Decay; Modeling Data
- Chapter 5: Systems of Equations and Inequalities
- Chapter 5.1: Systems of Linear Equations in Two Variables
- Chapter 5.2: Systems of Linear Equations in Three Variables
- Chapter 5.3: Partial Fractions
- Chapter 5.4: Systems of Nonlinear Equations in Two Variables
- Chapter 5.5: Systems of Inequalities
- Chapter 5.6: Linear Programming
- Chapter 6: Matrices and Determinants
- Chapter 6.1: Matrix Solutions to Linear Systems
- Chapter 6.2: Inconsistent and Dependent Systems and Their Applications
- Chapter 6.3: Matrix Operations and Their Applications
- Chapter 6.4: Multiplicative Inverses of Matrices and Matrix Equations
- Chapter 6.5: Determinants and Cramer's Rule
- Chapter 7: Conic Sections
- Chapter 7.1: The Ellipse
- Chapter 7.2: The Hyperbola
- Chapter 7.3: The Parabola
- Chapter 8: Sequences, Induction, and Probability
- Chapter 8.1: Sequences and Summation Notation
- Chapter 8.2: Arithmetic Sequences
- Chapter 8.3: Geometric Sequences and Series
- Chapter 8.4: Mathematical Induction
- Chapter 8.5: The Binomial Theorem
- Chapter 8.6: Counting Principles, Permutations, and Combinations
- Chapter 8.7: Probability
- Chapter P: Prerequisites: Fundamental Concepts of Algebra
- Chapter P.1: Algebraic Expressions, Mathematical Models, and Real Numbers
- Chapter P.2: Exponents and Scientific Notation
- Chapter P.3: Radicals and Rational Exponents
- Chapter P.4: Polynomials
- Chapter P.5: Factoring Polynomials
- Chapter P.6: Rational Expressions
College Algebra 6th Edition - Solutions by Chapter
Full solutions for College Algebra | 6th Edition
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
A sequence of steps intended to approach the desired solution.
= Xl (column 1) + ... + xn(column n) = combination of columns.
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Pseudoinverse A+ (Moore-Penrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.
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